If reputation and pride is what every warrior is after, then competition is the way to get it. People in the Iliad compete in just about everything. Excelling on the battlefield is the most obvious way to get street cred, followed by success in athletics. But people also compete over things like speaking ability, prizes, and, of course, political authority. One of the few times in the Iliad that someone wishes that someone else was better than them is when Hektor prays that his son will grow up to be a better warrior than him. On the one hand, this brings out Hektor's identity as a family man; on the other hand, it could just mean that he wants to be able to boast about having a better son than anyone else.
It is impossible to imagine the society of the Iliad without competition.
The Iliad shows respect for characters who succeed in one field even if they fail in others.